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Theorem bcthlem2 24849
Description: Lemma for bcth 24853. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)
Hypotheses
Ref Expression
bcth.2 𝐽 = (MetOpenβ€˜π·)
bcthlem.4 (πœ‘ β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
bcthlem.5 𝐹 = (π‘˜ ∈ β„•, 𝑧 ∈ (𝑋 Γ— ℝ+) ↦ {⟨π‘₯, π‘ŸβŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ∧ (π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))})
bcthlem.6 (πœ‘ β†’ 𝑀:β„•βŸΆ(Clsdβ€˜π½))
bcthlem.7 (πœ‘ β†’ 𝑅 ∈ ℝ+)
bcthlem.8 (πœ‘ β†’ 𝐢 ∈ 𝑋)
bcthlem.9 (πœ‘ β†’ 𝑔:β„•βŸΆ(𝑋 Γ— ℝ+))
bcthlem.10 (πœ‘ β†’ (π‘”β€˜1) = ⟨𝐢, π‘…βŸ©)
bcthlem.11 (πœ‘ β†’ βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)))
Assertion
Ref Expression
bcthlem2 (πœ‘ β†’ βˆ€π‘› ∈ β„• ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
Distinct variable groups:   π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   𝐢,π‘Ÿ,π‘₯   𝑔,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧,𝐷   𝑔,𝐹,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   𝑔,𝐽,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   𝑔,𝑀,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   πœ‘,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   π‘₯,𝑅   𝑔,𝑋,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧
Allowed substitution hints:   πœ‘(𝑔)   𝐢(𝑧,𝑔,π‘˜,𝑛)   𝑅(𝑧,𝑔,π‘˜,𝑛,π‘Ÿ)
Proof of Theorem bcthlem2
StepHypRef Expression
1 bcthlem.11 . . . . 5 (πœ‘ β†’ βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)))
2 fvoveq1 7434 . . . . . . 7 (π‘˜ = 𝑛 β†’ (π‘”β€˜(π‘˜ + 1)) = (π‘”β€˜(𝑛 + 1)))
3 id 22 . . . . . . . 8 (π‘˜ = 𝑛 β†’ π‘˜ = 𝑛)
4 fveq2 6891 . . . . . . . 8 (π‘˜ = 𝑛 β†’ (π‘”β€˜π‘˜) = (π‘”β€˜π‘›))
53, 4oveq12d 7429 . . . . . . 7 (π‘˜ = 𝑛 β†’ (π‘˜πΉ(π‘”β€˜π‘˜)) = (𝑛𝐹(π‘”β€˜π‘›)))
62, 5eleq12d 2827 . . . . . 6 (π‘˜ = 𝑛 β†’ ((π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)) ↔ (π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›))))
76rspccva 3611 . . . . 5 ((βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)) ∧ 𝑛 ∈ β„•) β†’ (π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)))
81, 7sylan 580 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)))
9 bcthlem.9 . . . . . 6 (πœ‘ β†’ 𝑔:β„•βŸΆ(𝑋 Γ— ℝ+))
109ffvelcdmda 7086 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (π‘”β€˜π‘›) ∈ (𝑋 Γ— ℝ+))
11 bcth.2 . . . . . . 7 𝐽 = (MetOpenβ€˜π·)
12 bcthlem.4 . . . . . . 7 (πœ‘ β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
13 bcthlem.5 . . . . . . 7 𝐹 = (π‘˜ ∈ β„•, 𝑧 ∈ (𝑋 Γ— ℝ+) ↦ {⟨π‘₯, π‘ŸβŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ∧ (π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))})
1411, 12, 13bcthlem1 24848 . . . . . 6 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ (π‘”β€˜π‘›) ∈ (𝑋 Γ— ℝ+))) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)) ↔ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)))))
1514expr 457 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((π‘”β€˜π‘›) ∈ (𝑋 Γ— ℝ+) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)) ↔ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))))))
1610, 15mpd 15 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)) ↔ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)))))
178, 16mpbid 231 . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))))
18 cmetmet 24810 . . . . . . . . . . . 12 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
1912, 18syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝐷 ∈ (Metβ€˜π‘‹))
20 metxmet 23847 . . . . . . . . . . 11 (𝐷 ∈ (Metβ€˜π‘‹) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
2119, 20syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
2211mopntop 23953 . . . . . . . . . 10 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2321, 22syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐽 ∈ Top)
24 xp1st 8009 . . . . . . . . . . . 12 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋)
25 xp2nd 8010 . . . . . . . . . . . . 13 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ+)
2625rpxrd 13019 . . . . . . . . . . . 12 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*)
2724, 26jca 512 . . . . . . . . . . 11 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋 ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*))
28 blssm 23931 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋 ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† 𝑋)
29283expb 1120 . . . . . . . . . . 11 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ ((1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋 ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*)) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† 𝑋)
3021, 27, 29syl2an 596 . . . . . . . . . 10 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† 𝑋)
31 df-ov 7414 . . . . . . . . . . . 12 ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) = ((ballβ€˜π·)β€˜βŸ¨(1st β€˜(π‘”β€˜(𝑛 + 1))), (2nd β€˜(π‘”β€˜(𝑛 + 1)))⟩)
32 1st2nd2 8016 . . . . . . . . . . . . 13 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (π‘”β€˜(𝑛 + 1)) = ⟨(1st β€˜(π‘”β€˜(𝑛 + 1))), (2nd β€˜(π‘”β€˜(𝑛 + 1)))⟩)
3332fveq2d 6895 . . . . . . . . . . . 12 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) = ((ballβ€˜π·)β€˜βŸ¨(1st β€˜(π‘”β€˜(𝑛 + 1))), (2nd β€˜(π‘”β€˜(𝑛 + 1)))⟩))
3431, 33eqtr4id 2791 . . . . . . . . . . 11 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) = ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))))
3534adantl 482 . . . . . . . . . 10 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) = ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))))
3611mopnuni 23954 . . . . . . . . . . . 12 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
3721, 36syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝑋 = βˆͺ 𝐽)
3837adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ 𝑋 = βˆͺ 𝐽)
3930, 35, 383sstr3d 4028 . . . . . . . . 9 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† βˆͺ 𝐽)
40 eqid 2732 . . . . . . . . . 10 βˆͺ 𝐽 = βˆͺ 𝐽
4140sscls 22567 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† βˆͺ 𝐽) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))))
4223, 39, 41syl2an2r 683 . . . . . . . 8 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))))
43 difss2 4133 . . . . . . . 8 (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
44 sstr2 3989 . . . . . . . 8 (((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
4542, 43, 44syl2im 40 . . . . . . 7 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
4645a1d 25 . . . . . 6 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))))
4746ex 413 . . . . 5 (πœ‘ β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))))
48473impd 1348 . . . 4 (πœ‘ β†’ (((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
4948adantr 481 . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
5017, 49mpd 15 . 2 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
5150ralrimiva 3146 1 (πœ‘ β†’ βˆ€π‘› ∈ β„• ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βˆ– cdif 3945   βŠ† wss 3948  βŸ¨cop 4634  βˆͺ cuni 4908   class class class wbr 5148  {copab 5210   Γ— cxp 5674  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413  1st c1st 7975  2nd c2nd 7976  1c1 11113   + caddc 11115  β„*cxr 11249   < clt 11250   / cdiv 11873  β„•cn 12214  β„+crp 12976  βˆžMetcxmet 20935  Metcmet 20936  ballcbl 20937  MetOpencmopn 20940  Topctop 22402  Clsdccld 22527  clsccl 22529  CMetccmet 24778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-n0 12475  df-z 12561  df-uz 12825  df-q 12935  df-rp 12977  df-xneg 13094  df-xadd 13095  df-xmul 13096  df-topgen 17391  df-psmet 20942  df-xmet 20943  df-met 20944  df-bl 20945  df-mopn 20946  df-top 22403  df-topon 22420  df-bases 22456  df-cld 22530  df-cls 22532  df-cmet 24781
This theorem is referenced by:  bcthlem3  24850  bcthlem4  24851
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